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1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
1st Québec-Ontario Workshop on Insurance Mathematics
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1st Québec-Ontario Workshop on Insurance Mathematics

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  • 1. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty One year uncertainty in claims reserving Arthur Charpentier Université Rennes 1 (& Université de Montréal) arthur.charpentier@univ-rennes1.fr http ://freakonometrics.blog.free.fr/ Workshop on Insurance Mathematics, Montreal, January 2011. 1
  • 2. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty Agenda of the talk• Formalizing claims reserving problem• From Mack to Merz & Wüthrich• From Mack (1993) to Merz & Wüthrich (2009)• Updating Poisson-ODP bootstrap technique one year ultimate China ladder Merz & Wüthrich (2008) Mack (1993) GLM+boostrap x Hacheleister & Stanard (1975) England & Verrall (1999) 2
  • 3. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty ‘one year horizon for the reserve risk’ 3
  • 4. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty ‘one year horizon for the reserve risk’ 4
  • 5. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty ‘one year horizon for the reserve risk’ 5
  • 6. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty ‘one year horizon for the reserve risk’ 6
  • 7. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty ‘one year horizon for the reserve risk’ 7
  • 8. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty1 Formalizing the claims reserving problem• Xi,j denotes incremental payments, with delay j, for claims occurred year i,• Ci,j denotes cumulated payments, with delay j, for claims occurred year i, Ci,j = Xi,0 + Xi,1 + · · · + Xi,j , 0 1 2 3 4 5 0 1 2 3 4 5 0 3209 1163 39 17 7 21 0 3209 4372 4411 4428 4435 4456 1 3367 1292 37 24 10 1 3367 4659 4696 4720 4730 2 3871 1474 53 22 et et 2 3871 5345 5398 5420 3 4239 1678 103 3 4239 5917 6020 4 4929 1865 4 4929 6794 5 5217 5 5217• Hn denotes information available at time n, Hn = {(Ci,j ), 0 ≤ i + j ≤ n} = {(Xi,j ), 0 ≤ i + j ≤ n}Actuaries have to predict the total amount of payments for accident year i, i.e. (n−i) Ci,n = E[Ci,∞ |Hn ] = E[Ci,n |Hn ]The difference betwwen what should be paid, and what has been paid will be the 8
  • 9. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty (n−i)claims reserve,Ri = Ci,n − Ci,n−i .A classical measure of uncertainty in claims reserving is mse[Ci,n |Fi,n−i ] calledultimate uncertainty.In Solvency II, it is now requiered to quantify one year uncertainy. The startingpoint is that in one year we will update our prediction (n−i+1) Ci,n = E[Ci,n |Hn+1 ]so that there is a change (compared with now) (n−i+1) (n−i) ∆n = CDRi,n = Ci,n i − Ci,n .If that difference is positive, the insurance will experience a mali (the insurer willpay more than what was expected), and a boni if the difference is negative. Notethat E[∆n |Hn ] = 0. In Solvency II, actuaries have to calculate uncertainty iassociated to ∆n , i.e. mse[∆n |Hn ]. i i 9
  • 10. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 10
  • 11. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 11
  • 12. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 12
  • 13. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 3500 4000 4500 5000 5500 6000 6500 Montant (paiements et réserves) 0 1 2 3 4 Figure 1 – Estimation of total payments Ci,n two consecutive years. 13
  • 14. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty2 Chain Ladder and claims reservingA standard approach is to assume that Ci,j+1 = λj · Ci,j for all i, j = 1, · · · , n.A natural estimator for λj , based on past experience is n−j i=1 Ci,j+1 λj = n−j for all j = 1, · · · , n − 1. i=1 Ci,jFuture payments can be predicted as follows, Ci,j = λn−i ...λj−1 Ci,n−i . 14
  • 15. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 0 1 2 3 4 5 λj 1,38093 1,01143 1,00434 1,00186 1,00474 1,0000 Table 1 – Development factors, λ = (λi ).Observe that n−j Ci,j Ci,j+1 λj = ωi,j λi,j où ωi,j = n−j et λi,j = . i=1 i=1 Ci,j Ci,ji.e. it is a weighted least squares estimator n−j 2 Ci,j+1 λj = argmin Ci,j λ− , λ∈R i=1 Ci,jor n−j 1 2 λj = argmin [λCi,j − Ci,j+1 ] . λ∈R i=1 Ci,j 15
  • 16. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 0 1 2 3 4 5 1 3209 4372 4411 4428 4435 4456 2 3367 4659 4696 4720 4730 4752.4 3 3871 5345 5398 5420 5430.1 5455.8 4 4239 5917 6020 6046.15 6057.4 6086.1 5 4929 6794 6871.7 6901.5 6914.3 6947.1 6 5217 7204.3 7286.7 7318.3 7331.9 7366.7Table 2 – Cumulated payments C = (Ci,j )i+j≤n and future projected paymentsC = (Ci,j )i+j>n , Ci,j = λj−1 · · · λn−i Ci,n−i . 16
  • 17. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyProposition1If there are A = (A0 , · · · , An ) and B = (B0 , · · · , Bn ), with B0 + · · · + Bn = 1, suchthat n−j n−j n−i n−i Ai B j = Yi,j for all j, and Ai B j = Yi,j for all i, i=1 i=1 j=0 j=0then n−1 Ci,n = Ai = Ci,n−i · λk k=n−iwhere n−1 n−1 n−1 1 1 1 Bk = − , avec B0 = . λj λj λj j=k j=k−1 j=kcf. margin method (Bailey (1963)), and Poisson regression. 17
  • 18. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty3 From Mack to Merz & Wüthrich3.1 Quantifying uncertaintyWe wish to compare R and R (where R is random - and unknown). The mse ofprediction can be written E([R − R]2 ) ≈ E([R − E(R)]2 ) + E([R − E(R)]2 ) mse(R) Var(R)with a first order approximation.Actually, what we are looking for is the msep conditional to the information wehave msepn (R) = E([R − R]2 |Hn ). 18
  • 19. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty3.2 Mack’s modelMack (1993) proposed a probabilistic model to justify Chain-Ladder. Assume 2that (Ci,j )j≥0 is a Markovian process, and there are λ = (λj ) and σ = (σj ) suchthat   E(C |H ) = E(C i,j+1 i+j |C ) = λ · C i,j+1 i,j j i,j  Var(Ci,j+1 |Hi+j ) = Var(Ci,j+1 |Ci,j ) = σ 2 · Ci,j jUnder those assumptions E(Ci,j+k |Hi+j ) = E(Ci,j+k |Ci,j ) = λj · λj+1 · · · λj+k−1 Ci,jMack (1993) assumed further that accident year are independent : (Ci,j )j=1,...,nand (Ci ,j )j=1,...,n are independent for all i = i .It is possible to write Ci,j+1 = λj Ci,j + σj Ci,j εi,j where residuals (εi,j ) arei.i.d., centred, with unit variance.=⇒ it is possible to used weigthed least squares, 19
  • 20. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty n−j 1 2 min (Ci,j+1 − λj Ci,j ) i=1 Ci,j n−j i=1 Ci,j+1 λj = n−j , ∀j = 1, · · · , n − 1. i=1 Ci,jis an unbiased estimator of λj .Further, a natural estimator for the variance parameter is n−j−1 2 2 1 Ci,j+1 − λj Ci,j σj = n−j−1 i=1 Ci,jwhich can be written n−j−1 2 2 1 Ci,j+1 σj = − λj · Ci,j n−j−1 i=1 Ci,j 20
  • 21. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty3.3 Uncertainty on Ri and RRecall that 2 mse(Ri ) = mse(Ci,n − Ci,n−i ) = mse(Ci,n ) = E Ci,n − Ci,n |HnProposition2The mean squared error of reserve estimate Ri mse(Ri ), for accident year i can beestimated by n−1 2 2 σk 1 1 mse(Ri ) = Ci,n + n−k . 2 Ci,k k=n−i λk j=1 Cj,kAnd the reserve all years R = R1 + · · · + Rn satisfies   n n 2 mse(R) = E  Ri − Ri |Hn  i=2 i=2Proposition3 21
  • 22. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyThe mean squared error of all year reserves mse(R), is given by n n−1 σk /λ2 2 k mse(R) = mse(Ri ) + 2 Ci,n Cj,n n−k . i=2 2≤i<j≤n k=n−i l=1 Cl,kExample1On our triangle mse(R) = 79.30, while mse(Rn ) = 68.45, mse(Rn−1 ) = 31.3 ormse(Rn−2 ) = 5.05. 22
  • 23. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty3.4 One year uncertainty, as in Merz & Wüthrich (2007) 0 1 2 3 4 1 3209 4372 4411 4428 4435 2 3367 4659 4696 4720 4727.4 3 3871 5345 5398 5422.3 5430.9 4 4239 5917 5970.0 5996.9 6006.4 5 4929 6810.8 6871.9 6902.9 6939.0Table 3 – Triangle of cumulated payments, seen as at year n = 5, C =(Ci,j )i+j≤n−1,i≤n−1 avec les projection future C = (Ci,j )i+j>n−1 . 23
  • 24. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 0 1 2 3 4 5 1 3209 4372 4411 4428 4435 4456 2 3367 4659 4696 4720 4730 4752.4 3 3871 5345 5398 5420 5430.1 5455.8 4 4239 5917 6020 6046.15 6057.4 6086.1 5 4929 6794 6871.7 6901.5 6914.3 6947.1Table 4 – Triangle of cumulated payments, seen as at year n = 6, on prior years,C = (Ci,j )i+j≤n,i≤n−1 avec les projection future C = (Ci,j )i+j>n . 24
  • 25. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyWe have to consider different transition factors n−i−1 n−i Ci,j+1 i=1 Ci,j+1 λn = j i=1 n−i−1 and λn+1 = j n−i i=1 Ci,j i=1 Ci,jWe know, e.g. that E(λn |Hn ) = λj and E(λn+1 |Hn+1 ) = λj .. But here, n + 1 is j jthe future, so we have to quantify E(λn+1 |Hn ). j nSet Sj = C1,j + C2,j + · · · , Cj,n−j , so that n−i n−i n−1−i i=1 Ci,j+1 i=1 Ci,j+1 Ci,j+1 Cn−j,j+1 λn+1 = j n−i = n+1 = i=1 n+1 + n+1 i=1 Ci,j Sj Sj Sji.e. Sj · λn n j Cn−j,j+1 λn+1 j = n+1 + n+1 . Sj Sj 25
  • 26. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyLemma1Under Mack’s assumptions n Sj Cn−j,n E(λn+1 |Hn ) = j n+1 · λn + λj · j n+1 . Sj SjNow let us defined the CDR,Definition1The claims development result CDRi (n + 1),for accident year i, between dates n andn + 1, is n n+1 CDRi (n + 1) = E(Ri |Hn ) − Yi,n−i+1 + E(Ri |Hn+1 ) ,where Yi,n−i+1 is the future incremental payment Yi,n−i+1 = Ci,n−i+1 − Ci,n−i .Note that CDRi (n + 1) is a Hn+1 -mesurable martingale, and CDRi (n + 1) = E(Ci,n |Hn ) − E(Ci,n |Hn+1 ). 26
  • 27. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyLemma2Under Mack’s assumption, an estimator for E(CDRi (n + 1)2 |Hn ) is 2 mse(CDRi (n + 1)|Hn ) = Ci,n Γi,n + ∆i,nwhere n−1 2 2 2 σn−i+1 Cn−j+1,j σj ∆i,n = n+1 + n+1 λ2 Sj n λ2 Sj n−i+1 Sn−i+1 j=n−i+2 jand 2 n−1 2 σn−i+1 σj Γi,n = 1+ 1+ n+1 Cn−j+1,j −1 λ2 n−i+1 Ci,n−i+1 j=n−i+2 λ2 [Sj ]2 jMerz & Wüthrich (2008) observed that n−1 2 2 2 σn−i+1 Cn−j+1,j σj Γi,n ≈ + n+1 λ2 n−i+1 Ci,n−i+1 j=n−i+2 Sj λ2 Cn−j+1,j j 27
  • 28. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintysince (1 + ui ) ≈ 1 + ui , if ui is small, i.e. 2 σj << Cn−j+1,j . λ2 jWe have finallyProposition4Under Mack’s assumptions n n [σn−i+1 ]2 1 1 msen (CDRi (n + 1)) ≈ [Ci,n ]2 + n [ λn n−i+1 ] 2 Ci,n−i+1 Sn−i+1   n−1 2 n [σj ]2 Cn−j+1,j +  1  . n 2 n n+1 j=n−i+2 [λj ] Sj Sjwhile Mack proposed n−1 n n n [σn−i+1 ]2 1 1 [σj ]2 1 1msen (Ri ) = [Ci,n ]2 + + + . n n [ λn n−i+1 ] 2 Ci,n−i+1 Sn−i+1 n 2 j=n−i+2 [λj ] Ci,j Sj 28
  • 29. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyi.e. only the first term of the model error is considered here, and only the first diagonal isconsidered for the process error (i + j = n + 1) (the other terms are neglected compared n+1with Cn−j+1,j /Sj ). 29
  • 30. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyFinally, all year, we have n msen (CDR(n + 1)) ≈ msen (CDRi (n + 1)) i=1   n−1 n n [σn−i ]2 /[λn ]2 n n−i Cn−j,j [σj ]2 /[λn ]2 n j +2 Ci,n Cl,n  i−1 + n−j n−j−1 . i<l k=0 Ck,n−i j=n−i+1 k=0 Ck,j k=0 Ck,j n+1again if Cn−j+1,j ≤ Sj .Example2On our triangle msen (CDR(n + 1)) = 72.57, while msen (CDRn (n + 1)) = 60.83,msen (CDRn−1 (n + 1)) = 30.92 or msen (CDRn−2 (n + 1)) = 4.48. La formuleapprochée donne des résultats semblables. 30
  • 31. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 31
  • 32. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty4 Poisson regressionWe have seen that a factor model Yi,j = ai × bj should be interesting (and can berelated to Chain Ladder)4.1 Hachemeister & StanardHachemeister & Stanard (1975), Kremer (1985) and Mack (1991) considered alog-Poisson regression on incremental payments E(Yi,j ) = µi,j = exp[ri + cj ] = ai · bjThen our best estimate is Yi,j = µi,j = exp[ri + cj ] = ai · bj .Example3On our triangle, we have 32
  • 33. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyCall:glm(formula = Y ~ lig + col, family = poisson("log"), data = base)Coefficients: Estimate Std. Error z value Pr(>|z|)(Intercept) 8.05697 0.01551 519.426 < 2e-16 ***lig2 0.06440 0.02090 3.081 0.00206 **lig3 0.20242 0.02025 9.995 < 2e-16 ***lig4 0.31175 0.01980 15.744 < 2e-16 ***lig5 0.44407 0.01933 22.971 < 2e-16 ***lig6 0.50271 0.02079 24.179 < 2e-16 ***col2 -0.96513 0.01359 -70.994 < 2e-16 ***col3 -4.14853 0.06613 -62.729 < 2e-16 ***col4 -5.10499 0.12632 -40.413 < 2e-16 ***col5 -5.94962 0.24279 -24.505 < 2e-16 ***col6 -5.01244 0.21877 -22.912 < 2e-16 ***--- 33
  • 34. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintySignif. codes: 0 ’***’ 0.001 ’**’ 0.01 ’*’ 0.05 ’.’ 0.1 ’ ’ 1(Dispersion parameter for poisson family taken to be 1) Null deviance: 46695.269 on 20 degrees of freedomResidual deviance: 30.214 on 10 degrees of freedom (15 observations deleted due to missingness)AIC: 209.52Number of Fisher Scoring iterations: 4 34
  • 35. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 0 1 2 3 4 5 1 3209 4372 4411 4428 4435 4456 2 3367 4659 4696 4720 4730 4752.4 3 3871 5345 5398 5420 5430.1 5455.8 4 4239 5917 6020 6046.15 6057.4 6086.1 5 4929 6794 6871.7 6901.5 6914.3 6947.1 6 5217 7204.3 7286.7 7318.3 7331.9 7366.7Table 5 – Triangle of cumulated payments, based on sums of Y = (Yi,j )0≤i,j≤n ’sobtained from the Poisson regression. 35
  • 36. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty4.2 Uncertainty in our regression model 8 q 6 4 q 2 q q q 0 0 2 4 6 8 10 36
  • 37. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty4.2.1 Les formules économétriques ferméesUsing standard GLM notions, Yi,j = µi,j = exp[ηi,j ].Using delta method we can write 2 ∂µi,j Var(Yi,j ) ≈ · Var(ηi,j ), ∂ηi,ji.e. (with a log link) ∂µi,j = µi,j ∂ηi,jIn an overdispersed Poisson regression (as in Renshaw (1998)), E [Yi,j − Yi,j ]2 ≈ φ · µi,j + µ2 · Var(ηi,j ) i,jfor the lower part of the triangle. Further, since Cov(Yi,j , Yk,l ) ≈ µi,j · µk,l · Cov(ηi,j , ηk,l ). 37
  • 38. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintywe get   E [R − R]2 ≈  φ · µi,j  + µ · Var(η) · µ i+j>nExample4On our triangle, the mean square error is 131.77 (to be compared with 79.30). 38
  • 39. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty4.2.2 Using bootstrap techniquesFrom triangle of incremental payments, (Yi,j ) assume that Yi,j ∼ P(Yi,j ) where Yi,j = exp(Li + Cj )1. Estimate parameters Li and Cj , define Pearson’s (pseudo) residuals Yi,j − Yi,j εi,j = Yi,j2. Generate pseudo triangles on the past, {i + j ≤ t} Yi,j = Yi,j + εi,j Yi,j3. (re)Estimate parameters Li and Cj , and derive expected payments for thefuture, Yi,j . R= Yi,j i+j>t 39
  • 40. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyis the best estimate.4. Generate a scenario for future payments, Yi,j e.g. from a Poisson distributionP(Yi,j ) R= Yi,j i+j>tOne needs to repeat steps 2-4 several times to derive a distribution for R. 40
  • 41. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 8000 q 6000 4000 2000 q q q q q 0 0 2 4 6 8 10 41
  • 42. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 8000 q 6000 4000 2000 q q q q q 0 0 2 4 6 8 10 42
  • 43. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 8000 q 6000 4000 2000 q q q q q 0 0 2 4 6 8 10 43
  • 44. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 8000 q q 6000 4000 2000 q q q q q 0 0 2 4 6 8 10 44
  • 45. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyIf we repeat it 50,000 times, we obtain the following distribution for the mse. 0.012 0.010 0.008 Density 0.006 0.004 MACK GLM 0.002 0.000 0 50 100 150 200 msep of overall reserves 45
  • 46. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty4.2.3 Bootstrap and one year uncertainty2. Generate pseudo triangles on the past and next year {i + j ≤ t + 1} Yi,j = Yi,j + εi,j Yi,j3. Estimate parameters Li and Cj , on the past, {i + j ≤ t}, and derive expectedpayments for the future, Yi,j . Rt = Yi,j i+j>t4. Estimate parameters Li and Cj , on the past and next year, {i + j ≤ t + 1},and derive expected payments for the future, Yi,j . Rt+1 = Yi,j i+j>t5. Calculate CDR as CDR=Rt+1 − Rt . 46
  • 47. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 8000 6000 4000 q 2000 q q q q q 0 0 2 4 6 8 10 47
  • 48. Arthur CHARPENTIER, WIM 2011, claims reserving uncertainty 8000 6000 4000 q 2000 q q q q q 0 0 2 4 6 8 10 48
  • 49. Arthur CHARPENTIER, WIM 2011, claims reserving uncertaintyultimate (R − E(R)) versus one year uncertainty, 0.010 0.008 0.006 0.004 0.002 0.000 −200 −100 0 100 200 49

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